110 XII! INTEGRATION
(13.4.4) Let (/zn) be an increasing sequence of real measures on X such that,
for each function /^ 0 belonging to JfR(X), the sequence (/-*„(/)) is bounded
above in R. Then the sequence (X,) has a vague limit in MR(X) which is also its
least upper bound for the order-relation on MR(X).
For each function /g: 0 in JTR(X), the sequence (jun(/)) is increasing and
bounded above in R, hence (4.2.1 ) has a limit //(/) in R which is equal to
sup jun(/). Since every function belonging to Jf C(X) is a linear combination
n
of four functions ^0 belonging to «#* R(X), it follows that the sequence (/in)
is vaguely convergent (13.4.1), and it is clear that \i is its least upper bound,
by the definition of the order-relation on MR(X).
(13.4.5) If a series of positive measures on X, with general term JUB, is such
that, for each f*z. 0 in jfR(X), the series with general term nn(f) «£ 0 is con-
vergent in R, then the series with general term \in is vaguely convergent in
MR(X), and its sum /i = £ /*„ is such that u(f) = £ Hn(f)for allfe Jf R(X).
n n
Apply (13.4.4) to the partial sums of the series.
PROBLEMS
1. Let A be Lebesgue measure on R, let JLG be its restriction to R* = ]0, + °o [, and let g
be the function x i—>• l/x on R* . Show that the measure g • jit cannot be extended to a
measure on R (cf. (17.9)).
2. On the real line R, show that the sequence of Dirac measures £„ (unit mass at the point
+TI) converges vaguely to 0. Give an example of a sequence (/,) in «2fc(R) which
converges to 0 in the Frechet space ^C(R) (12.14.6) but is such that the sequence
«£n>/n» tends to 1.
Let X be a compact space.
(a) Show that if (/„) is a sequence of functions in the Banach space ^C(X) which
tends to 0 and if (ju.n) is a sequence of measures on X which tends vaguely to 0, then
the sequence «/xn,/n» tends to 0 (use (13.4.2)).
(b) Suppose that X is infinite. Let V be a neighborhood of 0 in M(X) for the vague
topology, defined by a finite number, of inequalities | </*,/}>! ^ 1 (1 ^y ^ m)t and let
/be any function belonging to #C(X) which is not a linear combination of the/}.
Show that there exists ft e V such that | </x,/>| is arbitrarily large (cf. Section 12.15,
Problem 1). Deduce that the mapping (/A,/)i-> </*,/> of #c(X) x M(X) into C is
not continuous, and hence (using (a)) that the space M(X) is not metrizable with
respect to the vague topology.ch fi e H and eachfe «>f C(X), we have